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We present the detection and analysis of the phase curve of KELT-1b at optical wavelengths, analyzing data taken by the Transiting Exoplanet Survey Satellite (TESS). With a mass of ~27 M_J, KELT-1b is a low-mass brown dwarf. Due to the high mass and close proximity of its companion, the host star has a TESS light curve which shows clear ellipsoidal variations. We model the data with a six-component model: secondary eclipse, phase curve accounting for reflected light and thermal emission, Doppler beaming, ellipsoidal variations, stellar activity and the primary transit. We determine the secondary eclipse depth in the TESS bandpass to be 304 +/- 75 parts-per-million (ppm), the most accurate eclipse depth determined so far for KELT-1b. We measure the amplitude of the phase curve to be 128 +/- 27 ppm, with a corresponding eastward offset between the region of maximum brightness and the substellar point of 19.2 +/- 9.6 degrees, in good agreement with Spitzer measurements. We determine day and night brightness temperatures of 3201 +/- 147 K and 1484 +/- 110 K, respectively, slightly higher than those from Spitzer 3.6 and 4.5 micrometer data. A one-dimensional self-consistent atmospheric model can explain the TESS and Spitzer day side brightness temperatures with thermal emission alone and no reflected light. The night side data can be explained by a model with an internal temperature of ~1100 K, which may be related to the inflated radius. The difference between the TESS and Spitzer brightness temperatures can be explained by stronger molecular opacity in the Spitzer bands. On the night side, this opacity is due primarily to CH4 and CO while on the day side it is due to H2-H2 and H2-He collision-induced absorption.